Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
